Even with his ring laser idea, the fundamental problem is still the enormous gravitational field required to bend the space-time continuum in any meaningful way. It is the same gravitational survivability problem that exists with the "natural time machine" of a black hole.
Meanwhile, we have the same time travel option we have always had -- moving one reference frame at really high speed relative to another. This is what makes astronauts age just a little bit less during a mission than they would have if they had stayed on Earth. But as time travel goes, it's a one-way trip...

Think of Space/Time matrix or the Contiuum for short as the road. Think of the photon as the auto that follows the road. Gravity bends the road but since the photon has no rest mass is not effected by gravity but only follows the road. The key Dr. Mallett is to bend the road and the auto will follow. You are using the auto to bend the road. Use plasma to bend the road and the auto will follow. True warp of Time and Space. SpacedustSC

According to "sources" , those "GPS Scientists" knew in advance what the clock errors would be.
Essentially a built in correction factor was initiated and all was well.
At their altitude (distance from the earths gravitational center) the clocks ran ~7us faster per day.
Due to the velocity they ran ~45us slower per day.
Thus the total difference -- the satellite clocks were running 38 us per day slower relative to the earth clocks.
This was yet another affirmation of Einstein's theory. Ron R

It seems, for the moment, that one can only travel forward in time. You get on your rocket and fly around for a while, and when you land civilization has moved forward a few centuries. Not only does this avoid the paradox problem, but it pretty much eliminates cryogenics as the solution to medical issues. You fly around and return ... meanwhile your bank account grows ... if you don't get a cure, you take some of your money and fly around again ... meanwhile your bank account grows ... and so on. Your relatives don't get your money or your body.